Interval Methods and Taylor Model Methods for ODEs

Transcription

1 Interval Methods and Taylor Model Methods for ODEs Markus Neher, Dept. of Mathematics KARLSRUHE INSTITUTE OF TECHNOLOGY (KIT) 0 TM VII, Key West KIT University of the State of Baden-Wuerttemberg and Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT National Research Center of the Helmholtz Association

2 Validated Methods Also called: Verified Methods, Rigorous Methods, Guaranteed Methods, Enclosure Methods,... Aim: Compute guaranteed bounds for the solution of a problem, including Discretization errors (ODEs, PDEs, optimization) Truncation errors (Newton s method, summation) Roundoff errors Used for Modelling of uncertain data Bounding of roundoff errors 1 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

3 Outline 1 Interval arithmetic 2 Interval methods for ODEs 3 Taylor model methods for ODEs 2 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

4 Interval Arithmetic 3 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

5 Interval Arithmetic Compact real intervals: IR = {x = [x, x] x x} (x, x R). Basic arithmetic operations: x y := {x y x x, y y}, {+,,, /} (0 y for /) x + y = [x + y, x + y] x y = [x y, x y] x y = [min{xy, xy, xy, xy, }, max{xy, xy, xy, xy, }], x / y = x [1 / y, 1 / y] 4 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

6 Interval Arithmetic Compact real intervals: IR = {x = [x, x] x x} (x, x R). Basic arithmetic operations: x y := {x y x x, y y}, {+,,, /} (0 y for /) x + y = [x + y, x + y] x y = [x y, x y] [0, 1] [0, 1] = [ 1, 1] x y = [min{xy, xy, xy, xy, }, max{xy, xy, xy, xy, }], x / y = x [1 / y, 1 / y] 4 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

7 Interval Arithmetic Compact real intervals: IR = {x = [x, x] x x} (x, x R). Basic arithmetic operations: x y := {x y x x, y y}, {+,,, /} (0 y for /) x + y = [x + y, x + y] FPIA : [ x + y, x + y ] x y = [x y, x y] [0, 1] [0, 1] = [ 1, 1] x y = [min{xy, xy, xy, xy, }, max{xy, xy, xy, xy, }], x / y = x [1 / y, 1 / y] 4 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

8 Ranges and Inclusion Functions Range of f : D E: Rg (f, D) := {f (x) x D} Inclusion function F : IR IR of f : D R R: Examples: F(x) Rg (f, x) for all x D x 1 + x, 1 1, are inclusion functions for 1 + x f (x) = x 1 + x = x e x := [e x, e x ] is an inclusion function for e x 5 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

9 IA: Dependency f (x) = x 1 + x x 1 + x = 1 1, x = [1, 2]: 1 + x = [1, 2] [2, 3] = [ 1 3, 1] x = 1 1 [2, 3] = 1 [ 1 3, 1 2 ] = [ 1 2, 2 ] = Rg (f, x) 3 Reduced overestimation: centered forms, etc. Mean value form: Rg (f, x) f (c) + F (x)(x c), c = m(x). 6 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

10 IA: Wrapping Effect Overestimation: Enclose non-interval shaped sets by intervals 2 Example: f : (x, y) (x + y, y x) (Rotation) 2 Interval evaluation of f on x = ([ 1, 1], [ 1, 1]): y x 7 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

11 IA: Wrapping Effect Overestimation: Enclose non-interval shaped sets by intervals 2 Example: f : (x, y) (x + y, y x) (Rotation) 2 Interval evaluation of f on x = ([ 1, 1], [ 1, 1]): y y x x TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

12 Interval Methods for ODEs 8 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

13 Validated Integration of ODEs Interval IVP: u = f (t, u), u(t 0 ) = u 0 u 0, t t = [t 0, t end ] f : R R m R m sufficiently smooth, u 0 IR m, t end > t 0. 9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

14 Validated Integration of ODEs Interval IVP: u = f (t, u), u(t 0 ) = u 0 u 0, t t = [t 0, t end ] f : R R m R m sufficiently smooth, u 0 IR m, t end > t 0. 9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

15 Validated Integration of ODEs Interval IVP: u = f (t, u), u(t 0 ) = u 0 u 0, t t = [t 0, t end ] f : R R m R m sufficiently smooth, u 0 IR m, t end > t 0. 9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

16 Validated Integration of ODEs Interval IVP: u = f (t, u), u(t 0 ) = u 0 u 0, t t = [t 0, t end ] f : R R m R m sufficiently smooth, u 0 IR m, t end > t 0. 9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

17 Validated Integration of ODEs Interval IVP: u = f (t, u), u(t 0 ) = u 0 u 0, t t = [t 0, t end ] f : R R m R m sufficiently smooth, u 0 IR m, t end > t 0. 9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

18 Validated Integration of ODEs Interval IVP: u = f (t, u), u(t 0 ) = u 0 u 0, t t = [t 0, t end ] f : R R m R m sufficiently smooth, u 0 IR m, t end > t 0. 9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

19 Validated Integration of ODEs Interval IVP: u = f (t, u), u(t 0 ) = u 0 u 0, t t = [t 0, t end ] f : R R m R m sufficiently smooth, u 0 IR m, t end > t 0. 9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

20 Validated Integration of ODEs Interval IVP: u = f (t, u), u(t 0 ) = u 0 u 0, t t = [t 0, t end ] f : R R m R m sufficiently smooth, u 0 IR m, t end > t 0. 9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

21 Validated Integration of ODEs Interval IVP: u = f (t, u), u(t 0 ) = u 0 u 0, t t = [t 0, t end ] f : R R m R m sufficiently smooth, u 0 IR m, t end > t 0. 9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

22 Validated Integration of ODEs Interval IVP: u = f (t, u), u(t 0 ) = u 0 u 0, t t = [t 0, t end ] f : R R m R m sufficiently smooth, u 0 IR m, t end > t 0. 9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

23 Validated Integration of ODEs Interval IVP: u = f (t, u), u(t 0 ) = u 0 u 0, t t = [t 0, t end ] f : R R m R m sufficiently smooth, u 0 IR m, t end > t 0. 9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

24 Validated Integration of ODEs Interval IVP: u = f (t, u), u(t 0 ) = u 0 u 0, t t = [t 0, t end ] f : R R m R m sufficiently smooth, u 0 IR m, t end > t 0. 9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

25 Validated Integration of ODEs Interval IVP: u = f (t, u), u(t 0 ) = u 0 u 0, t t = [t 0, t end ] f : R R m R m sufficiently smooth, u 0 IR m, t end > t 0. 9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

26 Autonomous Interval IVP u = f (u), u(t 0 ) = u 0 u 0, t t = [t 0, t end ], where D R m, f C n (D), f : D R m, u 0 IR m. Moore s enclosure method: Automatic computation of Taylor coefficients Interval iteration: For j = 1, 2,... : A priori enclosure: v j u(t) for all t [t j 1, t j ] ("Alg. I"). Truncation error: z j := hn+1 j (n + 1)! f (n) (v j ). u(t j ) u j := u j 1 + n k=1 h k j k! f (k 1) (u j 1 ) + z j ("Algorithm II"). 10 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

27 Piecewise Constant A Priori Enclosure u v 1 v 2 u 0 u 1 u 2 t 0 t 1 t 2 t 11 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

28 A Priori Enclosures Picard iteration: find h j, v j such that u j 1 + [0, h j ]f (v j ) v j Step size restrictions: Explicit Euler steps Improvements: Lohner 1988, Corliss & Rihm 1996, Makino 1998, Nedialkov & Jackson 2001 Alternatives: Neumaier 1994, N. 1999, N. 2007, Delanoue & Jaulin 2010, Kin, Kim & Nakao TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

29 Modifications of Algorithm II Reduction of wrapping effect: Moore, Eijgenraam, Lohner, Rihm, Kuehn, Nedialkov & Jackson,... Nedialkov & Jackson: Hermite-Obreshkov-Method Rihm: Implicit methods Petras & Hartmann, Bouissou: Runge-Kutta-Methods Berz & Makino: Taylor models Taylor expansion of solution w.r.t. time and initial values 13 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

30 Direct Interval Method Direct method (Moore 1965): Apply mean value form to f (k) : ( ) f [0] (u) = u, f [k] (u) = 1 k f [k 1] u f (u) for k 1. Let S j 1 = I + n k=1 h k 0 J( f [k] (u j 1 ) ), z j = h n+1 0 f [n] (v j ), (I: identity matrix, J ( f [k]) : Jacobian of f [i] ), then for some û j 1 u j 1 n 1 u(t j ; u 0 ) u j = û j 1 + hj k f [k] (û j 1 ) + z j + S j 1 (u j 1 û j 1 ). k=1 14 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

31 Global Error Propagation Wrapping effect: S j 1 (u j 1 û j 1 ) may overestimate S = {S j 1 (u j 1 û j 1 ) S j 1 S j 1, u j 1 u j 1 } propagate S as a parallelepiped. û 0 := m(u 0 ), r 0 = u 0 û 0, B 0 = I; for some nonsingular B j 1 : n 1 û j = û j 1 + hj 1 k f [k] (û j 1 ) + m(z j ), u j = û j 1 + k=1 n 1 k=1 h k j 1 f [k] (û j 1 ) + z j + (S j 1 B j 1 )r j 1, û j : approximate point solution for the central IVP z j : local error; r j : global error 15 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

32 Global Error Propagation Global error propagation: ( ) r j = B 1 j (S j 1 B j 1 ) r j 1 + B 1 j (z j m(z j )) Moore s direct method: B j = I Pep method (Eijgenraam, Lohner): B j = m(s j 1 B j 1 ) QR method (Lohner): m(s j 1 B j 1 ) = QR, B j := Q Blunting method (Berz, Makino): modify B j in the pep method such that condition numbers remain small 16 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

33 Autonomous Linear ODE Autonomous linear system (A R m m ): u = A u, u(0) u 0. Propagation of the Global Error r j = (B 1 j TB j 1 )r j 1 + B 1 ( zj m(z j ) ), T = j n 1 ν=0 (ha) ν. ν! + = 17 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

34 Interval Methods for Linear ODEs Direct method: r j = T r j 1 + z j m(z j ). Optimal for local error, bad for global error (rotation) Parallelepiped method: r j = r j 1 + (T j )(z j (m(z j )). Optimal for global error; suitable for local error, if cond(t j ) is small Bad for local error in presence of shear QR method: r j = R j r j 1 + Qj T ( zj m(z j ) ), j = 1, 2,.... Handles rotation, contraction, shear 18 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

35 Wrapping Effect: Direct Interval Method (Plots by Ned Nedialkov) Huge overestimations in general 19 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

36 Wrapping Effect: Parallelepiped Method B j = TB j 1 3 x x { C j r +e r r j 1, e e j } { C j r r r j 1 } 4 { C j r +e r r j 1, e e j } { C j r r r j 1 } { C 3 j r r r j } x { C j r r r j } x B j often ill-conditioned, large overestimations 20 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

37 Wrapping Effect: QR Method B j = Q j, Q j R j = TB j 1 x x { C j r +e r r j 1, e e j } 1.5 { C j r +e r r j 1, e e j } 3 { C j r r r j 1 } 2 { C j r r r j 1 } 4 QR, Version 1 QR, Version QR, Version 1 QR, Version x x Overestimation depends on column permutations of B j 1 21 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

38 Taylor Model Methods for ODEs 22 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

39 Taylor Models (I) x R m, f : x R, f C n+1, x 0 x; f (x) = p n,f (x x 0 ) + R n,f (x x 0 ), x x (p n,f Taylor polynomial, R n,f remainder term) Interval remainder bound of order n of f on x: x x : R n,f (x x 0 ) i n,f Taylor model T n,f = (p n,f, i n,f ) of order n of f : x x: f (x) p n,f (x x 0 ) + i n,f 23 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

40 Taylor Models (II) Taylor model: U := p n (x) + i, x x, x IR m, i IR m (p n : vector of m-variate polynomials of order n) Function set: U = {f C 0 (x) : f (x) p n (x) + i for all x x } Range of a TM: Rg (U) = {z = p(x) + ξ x x, ξ i} R m ( ) x Ex.: U := x1 2 + x, x 1, x 2 [ 1, 1] 2 x 2 Rg (U): x 1 24 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

41 TM Arithmetic Paradigm for TMA: p n,f is processed symbolically to order n Higher order terms are enclosed into the remainder interval of the result Taylor Model Methods for ODEs Taylor expansion of solution w.r.t. time and initial values Computation of Taylor coefficients by Picard iteration: Parameters describing initial set treated symbolically Interval remainder bounds by fixed point iteration (Makino, 1998) 25 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

42 Example: Quadratic Problem u = v, u(0) [0.95, 1.05], v = u 2, v(0) [ 1.05, 0.95]. Taylor model method: initial set described by parameters a and b: u 0 (a, b) := 1 + a, a a := [ 0.05, 0.05], v 0 (a, b) := 1 + b, b b := [ 0.05, 0.05]. 26 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

43 3rd order TM Method: Enclosure of the Flow h = 0.1, flow for τ [0, 0.1]: Ũ 1 (τ, a, b) := 1 + a τ + bτ τ2 + aτ τ3 + i 0, Ṽ 1 (τ, a, b) := 1 + b + τ + 2aτ τ 2 + a 2 τ aτ 2 + bτ τ3 + j 0. Flow at t 1 = 0.1: U 1 (a, b) := Ũ 1 (0.1, a, b) = a + 0.1b + i 0, V 1 (a, b) := Ṽ 1 (0.1, a, b) = a b + 0.1a 2 + j 0 (nonlinear boundary). 27 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

44 Naive TM Method Interval remainder terms accumulate Linear ODEs: Naive TM method performs similarly to the direct interval method Shrink wrapping, preconditioned TM methods 28 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

45 Shrink Wrapping Absorb interval term into polynomial part (Makino and Berz 2002): ( U V ) (white) vs. ( Usw V sw ). Linear ODEs: Shrink wrapping performs similarly to the pep method. 29 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

46 Integration with Preconditioned Taylor Models Preconditioned integration: flow at t j : U j = U l,j U r,j = (p l,j + i l,j ) (p r,j + i r,j ). Purpose: stabilize integration as in the QR interval method Theorem (Makino and Berz 2004) If the initial set of an IVP is given by a preconditioned Taylor model, then integrating the flow of the ODE only acts on the left Taylor model. 30 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

47 Preconditioned TMM for linear ODE Global error: i r,j+1 := C 1 l,j+1 TC l,j i r,j + C 1 l,j+1 i l,j+1, j = 0, 1,.... C l,j+1 = TC l,j : C l,j+1 = Q j : parallelepiped preconditioning QR preconditioning Other choices: curvilinear coordinates, blunting (Makino and Berz 2004) 31 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

48 Integration of Quadratic Problem u = v, u(0) [0.95, 1.05], v = u 2, v(0) [ 1.05, 0.95] COSY Infinity AWA 32 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT

49 Summary Interval arithmetic Interval methods for ODEs Taylor model methods for ODEs Open problems Treatment of high-dimensional problems Validated implicit methods 33 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT